Monte Carlo Variance Reduction Techniques

C++ implementations of variance reduction methods for pricing financial options using Monte Carlo simulation.

Methods Implemented

1. Antithetic Variates

Uses paired samples (Z, -Z) to exploit negative correlation and reduce variance.

• Variance Reduction Factor observed: 1.25x

2. Control Variates

Adjusts estimates using a correlated variable with known mean.

• Variance Reduction Factor observed: 7.72x

3. Stratified Sampling

Partitions probability space into strata and samples from each.

• Variance Reduction Factor observed: 11.1x

4. Latin Hypercube Sampling (LHS)

Multi-dimensional stratification with better space-filling properties.

• Variance Reduction Factor observed: 1.6x

5. Importance Sampling

Samples from shifted distribution centered near important regions.

• Variance Reduction Factor observed: 7.3x

Quick Start

Compile:

g++ -o method method_name.cpp

Run:

./method

Key Formulas

European Call Option: $$C(0) = e^{-rT} \mathbb{E}[\max(S(T) - K, 0)]$$

$$S(T) = S_0 \exp\left(\left(r - \frac{1}{2}\sigma^2\right)T + \sigma\sqrt{T}Z\right)$$

Variance Reduction Factor:

VRF = Var(Standard MC) / Var(Variance Reduction Method)

References

• Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering Chapter 4: Variance Reduction Techniques